Why: Help us understand which patient- or drug-specific characteristics are important determinants of PK parameters.
Mechanistic understanding:
Impact of disease type on CL.
Drug-drug interactions.
Controlling PK variability:
Dose selection for a special population (e.g., obese patients).
Dosing interval adjustment for extended release formulation.
How: Specify a mathematical relationship between a PK parameter and covariates.
Usually requires adding parameters (mostly THETAs) in the model.
Continuous covariates
Example: Body weight (WT) on CL
Linear:
CL=THETA(1)*(1+THETA(2)*(WT-70))
Piece-wise (hockey-stick) linear:
IF(WT.LE.50) CL=THETA(1)*(1+THETA(2)*(WT-70))
IF(WT.GT.50) CL=THETA(1)*(1+THETA(3)*(WT-70))
Exponential:
CL=THETA(1)*EXP(THETA(2)*(WT-70))
Power:
CL=THETA(1)*(WT/70)**THETA(2)
Centering: in all of the above case, when WT=70, model collapse to CL=THETA(1)
Interpretation of THETA(1): Typical CL for a subject with 70 kg body weight.
If replacing 70 with 60, for example, the interpretation of THETA(1) changes to “typical CL for a subject with 60 kg body weight”, with no impact on model fittings.
Categorical covariates
Example: Sex (Male=1; Female=0)
Linear:
CL=THETA(1)
IF(SEX.EQ.1) CL=THETA(1)*(1+THETA(2))
Covariate modeling approaches
Selection methods:
Use the observed data to identify covariates based on statistical significance
Examples:
Step-wise covariate modeling (SCM), or similar methods (e.g., COSSAC, SAMBA, etc).
Machine learning methods: LASSO, tree-based methods, etc.
Pre-defined methods:
Pre-define covariates of interest before looking at the data.
Adding covariate based on
Clinical relevance: Adding a transporter effect on CL based on mechanistic understanding despite small effect size.
Regulartory interest: Adding Asian population effect as covariate to support a PMDA submission.
Ignoring statistical significance during modeling.
Model inference relies on post-modeling simulations (e.g., forest plots).
Examples:
Full-fixed effect modeling (FFEM)
Full random effect modeling (FREM)
Covariate modeling approaches
Active area of interest.
No single method is universally accepted as the “gold-standard”.
Choice of method depends on modeling objectives and context (i.e., fit-for-purpose)
Covariate examinations: BEFORE you start covariate modeling
Know you covariates
Missing values?
No missing values-Perfect!
Small proportion (5%-10%):
Impute with median/mode.
Multiple imputation with chain equations (MICEs).
Large proportion (e.g., 30%)
Treat “missing” as a separate category.
Range and distribution of continuous covariates (e.g., body weight).
Distribution wide enough?
Count in each category of categorical covariates (e.g., sex)
Imbalance across category? Male:Female=20:1
Data reduction: BEFORE you start covariate modeling
Examine covariate distributions for correlation/colinearity
Select covariates carrying unique information (relatively independent).
Rule of thumb: be cautious when \(R^2>0.3\).
Exclude or combine covariates carrying repetitive information.
WT, HT, BSA and BMI are highly correlated. If one included as a covariate, others may not need to be included.
BLACK (No=0, Yes=1), WHITE (No=0, Yes=1) and ASIAN (No=0, Yes=1). May lump into one covariate (BLACK=1, WHITE=2, ASIAN=3, OTHER=4).
Other consideration points
Was the study designed to estimate a covariate effect?
SNP impact on CL.
Inclusion criteria of a clinical study may impact the choice of covariates.
A study enroll only female patients (sex may be difficult to be explored as a covariate).
Prior knowledge (very subjective).
Eye colors on CL?
Clinical interest of covariates
Selection methods-(SCM)
Forward selection (univariate)
Base model -> Add one covariate -> Significantly better? -> Yes, selected
Base model -> Add another covariate -> Significantly better? -> No, not selected
…
Backward Elimination
Full model1 -> remove one covariate -> significantly worse?2 -> Yes, selected
Full model -> remove another covariate -> significantly worse? -> No, not selected
…
How to test significance?
Likelihood Ratio Test (LRT) for nesting models
Nesting example
Model 1:
TVCL = THETA(1)
Model 2:
TVCL = THETA(1)*(WT/70)**THETA(2)
If WT/70 equals 1, then (WT/70)**THETA(2) equals 1, resulting in TVCL = THETA(1).
So Model 1 is nested within Model 2.
Another nesting example
Model 1:
TVCL = THETA(1)
Model 2:
TVCL = THETA(1)IF(SEX.EQ.0)TVCL = TVCL * THETA(2)
With THETA(2) fixed to 1.0 (it’s null value)
TVCL = THETA(1) (if male)
TVCL = THETA(1) * 1 (if female)
No predictive difference between males and females when THETA(2) is set to its null value
Smaller model (without SEX) is nested
Nesting?
Model 1:
TVCL = THETA(1)
Model 2:
TVCL = THETA(1)*WT
No biological meaningful WT reference value to remove the effect of WT.
Be careful. You can add COV effect without nesting.
Statistical significance vs. clinical relevance
Example 1:
TVCL = THETA(1)IF(SEX.EQ.0)TVCL = TVCL*THETA(2)
Fix THETA(2)=1, OFV=69.9
Estimate THETA(2), THETA(2)estimated as 1.1, OFV=62.1
Clinical relevant?
Females have a 10% faster CL than males
Statistical significant?
dOFV=69.9-62.1=7.8
Using a P value criteria (0.01) with 1 DF, if dOFV>6.63, the COV is considered statistically significant.